One half of the book is authored by Yuval Fisher himself, while articles from another 12 experts in the field present material from different points of view. The focus here is solely on fractal image encoding, with the aim of providing a working code that is usable in applications, while containing the complete details of how to encode and decode images. An indispensable "how to" guide, combining the very latest results in the field. Of interest to a very wide audience, ranging from experts in image processing to high school students.
One half of the book is authored by Yuval Fisher himself, while articles from another 12 experts in the field present material from different points of view. The focus here is solely on fractal image encoding, with the aim of providing a working code that is usable in applications, while containing the complete details of how to encode and decode images. An indispensable "how to" guide, combining the very latest results in the field. Of interest to a very wide audience, ranging from experts in image processing to high school students.
1 Introduction.- 1.1 What Is Fractal Image Compression?.- 1.2 Self-Similarity in Images.- 1.3 A Special Copying Machine.- 1.4 Encoding Images.- 1.5 Ways to Partition Images.- 1.6 Implementation.- 1.7 Conclusion.- 2 Mathematical Background.- 2.1 Fractals.- 2.2 Iterated Function Systems.- 2.3 Recurrent Iterated Function Systems.- 2.4 Image Models.- 2.5 Affine Transformations.- 2.6 Partitioned Iterated Function Systems.- 2.7 Encoding Images.- 2.8 Other Models.- 3 Fractal Image Compression with Quadtrees.- 3.1 Encoding.- 3.2 Decoding.- 3.3 Sample Results.- 3.4 Remarks.- 3.5 Conclusion.- 4 Archetype Classification in an Iterated Transformation Image Compression Algorithm.- 4.1 Archetype Classification.- 4.2 Results.- 4.3 Discussion.- 5 Hierarchical Interpretation of Fractal Image Coding and Its Applications.- 5.1 Formulation of PIFS Coding/Decoding.- 5.2 Hierarchical Interpretation.- 5.3 Matrix Description of the PIFS Transformation.- 5.4 Fast Decoding.- 5.5 Super-resolution.- 5.6 Different Sampling Methods.- 5.7 Conclusions.- A Proof of Theorem 5.1 (Zoom).- B Proof of Theorem 5.2 (PIFS Embedded Function).- C Proof of Theorem 5.3 (Fractal Dimension of the PIFS Embedded Function).- 6 Fractal Encoding with HV Partitions.- 6.1 The Encoding Method.- 6.2 Efficient Storage.- 6.3 Decoding.- 6.4 Results.- 6.5 More Discussion.- 6.6 Other Work.- 7 A Discrete Framework for Fractal Signal Modeling.- 7.1 Sampled Signals, Pieces, and Piecewise Self-transformability.- 7.2 Self-transformable Objects and Fractal Coding.- 7.3 Eventual Contractivity and Collage Theorems.- 7.4 Affine Transforms.- 7.5 Computation of Contractivity Factors.- 7.6 A Least-squares Method.- 7.7 Conclusion.- A Derivation of Equation (7.9).- 8 A Class of Fractal Image Coders with Fast Decoder Convergence.- 8.1 Affine Mappings on Finite-Dimensional Signals.- 8.2 Conditions for Decoder Convergence.- 8.3 Improving Decoder Convergence.- 8.4 Collage Optimization Revisited.- 8.5 A Generalized Sufficient Condition for Fast Decoding.- 8.6 An Image Example.- 8.7 Conclusion.- 9 Fast Attractor Image Encoding by Adaptive Codebook Clustering.- 9.1 Notation and Problem Statement.- 9.2 Complexity Reduction in the Encoding Step.- 9.3 How to Choose a Block.- 9.4 Initialization.- 9.5 Two Methods for Computing Cluster Centers.- 9.6 Selecting the Number of Clusters.- 9.7 Experimental Results.- 9.8 Possible Improvements.- 9.9 Conclusion.- 10 Orthogonal Basis IFS.- 10.1 Orthonormal Basis Approach.- 10.2 Quantization.- 10.3 Construction of Coders.- 10.4 Comparison of Results.- 10.5 Conclusion.- 11 A Convergence Model.- 11.1 The r Operator.- 11.2 Lp Convergence of the RIFS Model.- 11.3 Almost Everywhere Convergence.- 11.4 Decoding by Matrix Inversion.- 12 Least-Squares Block Coding by Fractal Functions.- 12.1 Fractal Functions.- 12.2 Least-Squares Approximation.- 12.3 Construction of Fractal Approximation.- 12.4 Conclusion.- 13 Inference Algorithms for WFA and Image Compression.- 13.1 Images and Weighted Finite Automata.- 13.2 The Inference Algorithm for WFA.- 13.3 A Fast Decoding Algorithm for WFA.- 13.4 A Recursive Inference Algorithm for WFA.- A Sample Code.- A.l The Enc Manual Page.- A.2 The Dec Manual Page.- A.3 Enc.c.- A.4 Dec.c.- A.5 The Encoding Program.- A.6 The Decoding Program.- A.7 Possible Modifications.- B Exercises.- C Projects.- C.1 Decoding by Matrix Inversion.- C.2 Linear Combinations of Domains.- C.3 Postprocessing: Overlapping, Weighted Ranges, and Tilt.- C.4 Encoding Optimization.- C.5 Theoretical Modeling for Continuous Images.- C.6 Scan-line Fractal Encoding.- C.7 Video Encoding.- C.8 Single Encoding of Several Frames.- C.9 Edge-based Partitioning.- C.10 Classification Schemes.- C.l1 From Classification to Multi-dimensional Keys.- C.12 Polygonal Partitioning305.- C.13 Decoding by Pixel Chasing.- C.14 Second Iterate Collaging.- C.15 Rectangular IFS Partitioning.- C.16 Hexagonal Partitioning.- C.17 Parallel Processing.- C.18 Non-contractive IFSs.- D Comparison of Results.- E Original Images.
1 Introduction.- 1.1 What Is Fractal Image Compression?.- 1.2 Self-Similarity in Images.- 1.3 A Special Copying Machine.- 1.4 Encoding Images.- 1.5 Ways to Partition Images.- 1.6 Implementation.- 1.7 Conclusion.- 2 Mathematical Background.- 2.1 Fractals.- 2.2 Iterated Function Systems.- 2.3 Recurrent Iterated Function Systems.- 2.4 Image Models.- 2.5 Affine Transformations.- 2.6 Partitioned Iterated Function Systems.- 2.7 Encoding Images.- 2.8 Other Models.- 3 Fractal Image Compression with Quadtrees.- 3.1 Encoding.- 3.2 Decoding.- 3.3 Sample Results.- 3.4 Remarks.- 3.5 Conclusion.- 4 Archetype Classification in an Iterated Transformation Image Compression Algorithm.- 4.1 Archetype Classification.- 4.2 Results.- 4.3 Discussion.- 5 Hierarchical Interpretation of Fractal Image Coding and Its Applications.- 5.1 Formulation of PIFS Coding/Decoding.- 5.2 Hierarchical Interpretation.- 5.3 Matrix Description of the PIFS Transformation.- 5.4 Fast Decoding.- 5.5 Super-resolution.- 5.6 Different Sampling Methods.- 5.7 Conclusions.- A Proof of Theorem 5.1 (Zoom).- B Proof of Theorem 5.2 (PIFS Embedded Function).- C Proof of Theorem 5.3 (Fractal Dimension of the PIFS Embedded Function).- 6 Fractal Encoding with HV Partitions.- 6.1 The Encoding Method.- 6.2 Efficient Storage.- 6.3 Decoding.- 6.4 Results.- 6.5 More Discussion.- 6.6 Other Work.- 7 A Discrete Framework for Fractal Signal Modeling.- 7.1 Sampled Signals, Pieces, and Piecewise Self-transformability.- 7.2 Self-transformable Objects and Fractal Coding.- 7.3 Eventual Contractivity and Collage Theorems.- 7.4 Affine Transforms.- 7.5 Computation of Contractivity Factors.- 7.6 A Least-squares Method.- 7.7 Conclusion.- A Derivation of Equation (7.9).- 8 A Class of Fractal Image Coders with Fast Decoder Convergence.- 8.1 Affine Mappings on Finite-Dimensional Signals.- 8.2 Conditions for Decoder Convergence.- 8.3 Improving Decoder Convergence.- 8.4 Collage Optimization Revisited.- 8.5 A Generalized Sufficient Condition for Fast Decoding.- 8.6 An Image Example.- 8.7 Conclusion.- 9 Fast Attractor Image Encoding by Adaptive Codebook Clustering.- 9.1 Notation and Problem Statement.- 9.2 Complexity Reduction in the Encoding Step.- 9.3 How to Choose a Block.- 9.4 Initialization.- 9.5 Two Methods for Computing Cluster Centers.- 9.6 Selecting the Number of Clusters.- 9.7 Experimental Results.- 9.8 Possible Improvements.- 9.9 Conclusion.- 10 Orthogonal Basis IFS.- 10.1 Orthonormal Basis Approach.- 10.2 Quantization.- 10.3 Construction of Coders.- 10.4 Comparison of Results.- 10.5 Conclusion.- 11 A Convergence Model.- 11.1 The r Operator.- 11.2 Lp Convergence of the RIFS Model.- 11.3 Almost Everywhere Convergence.- 11.4 Decoding by Matrix Inversion.- 12 Least-Squares Block Coding by Fractal Functions.- 12.1 Fractal Functions.- 12.2 Least-Squares Approximation.- 12.3 Construction of Fractal Approximation.- 12.4 Conclusion.- 13 Inference Algorithms for WFA and Image Compression.- 13.1 Images and Weighted Finite Automata.- 13.2 The Inference Algorithm for WFA.- 13.3 A Fast Decoding Algorithm for WFA.- 13.4 A Recursive Inference Algorithm for WFA.- A Sample Code.- A.l The Enc Manual Page.- A.2 The Dec Manual Page.- A.3 Enc.c.- A.4 Dec.c.- A.5 The Encoding Program.- A.6 The Decoding Program.- A.7 Possible Modifications.- B Exercises.- C Projects.- C.1 Decoding by Matrix Inversion.- C.2 Linear Combinations of Domains.- C.3 Postprocessing: Overlapping, Weighted Ranges, and Tilt.- C.4 Encoding Optimization.- C.5 Theoretical Modeling for Continuous Images.- C.6 Scan-line Fractal Encoding.- C.7 Video Encoding.- C.8 Single Encoding of Several Frames.- C.9 Edge-based Partitioning.- C.10 Classification Schemes.- C.l1 From Classification to Multi-dimensional Keys.- C.12 Polygonal Partitioning305.- C.13 Decoding by Pixel Chasing.- C.14 Second Iterate Collaging.- C.15 Rectangular IFS Partitioning.- C.16 Hexagonal Partitioning.- C.17 Parallel Processing.- C.18 Non-contractive IFSs.- D Comparison of Results.- E Original Images.
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"This book will be of great value to anyone interested in fractal image compression... It is certainly the best description of Fractal Image Compression to date, and should be the reference in this field for several years to come." J.H. Hubbard, Professor of Mathematics, Cornell University
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